Genus–degree formula

In classical algebraic geometry, the genus–degree formula relates the degree d of an irreducible plane curve $C$ with its arithmetic genus g via the formula:

$g=\frac12 (d-1)(d-2).$

Here "plane curve" means that $C$ is a closed curve in the projective plane $\mathbb{P}^2$. If the curve is non-singular the geometric genus and the arithmetic genus are equal, but if the curve is singular, with only ordinary singularities, the geometric genus is smaller. More precisely, an ordinary singularity of multiplicity r decreases the genus by $\scriptstyle \frac12 r(r-1)$.

Proof

The proof follows immediately from the adjunction formula. For a classical proof see the book of Arbarello, Cornalba, Griffiths and Harris.

Generalization

For a non-singular hypersurface $H$ of degree d in the projective space $\mathbb{P}^n$ of arithmetic genus g the formula becomes:

$g=\binom{d-1}{n} , \,$

where $\tbinom{d-1}{n}$ is the binomial coefficient.

References

• This article incorporates material from the Citizendium article "Genus degree formula", which is licensed under the Creative Commons Attribution-ShareAlike 3.0 Unported License but not under the GFDL.
• Arbarello, Cornalba, Griffiths, Harris. Geometry of algebraic curves. vol 1 Springer, ISBN 0-387-90997-4, appendix A.
• Griffiths and Harris, Principles of algebraic geometry, Wiley, ISBN 0-471-05059-8, chapter 2, section 1.
• Robin Hartshorne (1977): Algebraic geometry, Springer, ISBN 0-387-90244-9.